Which of the following is a verbal version of the Product Law (assuming the limits exist)? (a) The product of two functions has a limit. (b) The limit of the products the product of the limits. (c) The product of a limitis a product of functions. (d) A limit produces a product of functions.

Question

Which of the following is a verbal version of the Product Law (assuming the limits exist)?    (a) The product of two functions has a limit.    (b) The limit of the products the product of the limits.    (c) The product of a limitis a product of functions.    (d) A limit produces a product of functions.

Tuthornt · Accepted Answer

Step1 To Determine: which of the following is a verbal version of the product law Given:we have two functions f and g Explanation: the option b is correct that is the limit of the product is the product proof: let we have two functions f and g such that limx→af(x)=l and limx→ag(x)=m to prove:limx→a(fg)(x)=lm let ∈&amp;gt;0 be given now |(fg)(x)−lm|=|f(x)g(x)−lg(x)+lg(x)−lm|≤|f(x)g(x)−lg(x)|+|lg(x)−lm| =g(x)|f(x)−l|+|l||g(x)−m| Step 2 since limx→ag(x)=m, therefore g(x) is bounded in some deleted neighbourhood of x=a hence there exist k&amp;gt;0 and δ1&amp;gt;0 such that |g(x)|≤k whenever 0&amp;lt;|x−a|&amp;lt;δ1 limx→af(x)=l and limx→ag(x)=m therefore, corresponding to any given ∈&amp;gt;0, we can find positive number δ2 and δ3 such that |f(x)−l|&amp;lt;∈2κ whenever 0&amp;lt;|x−a|&amp;lt;δ2

Which of the following is a verbal version of the

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